Local features such as lines, edges, and corners are the elements of form vision. Segmenting an image and estimating its surface properties depend on analyzing the spatial statistics of these components. Thus, visual textures constructed from these features constitute an important "perceptual space," in which distances correspond to the degree to which a texture difference supports the inference of a border. Here, we characterize basic aspects of the global geometry of this perceptual space. To reduce the dimensionality of the space to a practical level, we consider binary images and parameterize them by the configurations present in 2x2 neighborhoods, a strategy that focuses on image statistics that are informative in natural images (Tkačik et al., 2010). This leads to a 10-dimensional space of synthetic images, which constitutes a perceptual space, analogous to the familiar 3-dimensional color space. Our previous discrimination-threshold measurements (Victor et al., VSS 2013) showed that perceptual distances near the origin of this space corresponds to a Euclidean metric, and hence, a locally flat geometry. Here, we use suprathreshold border salience judgments to probe the global geometry of this space. Subjects (N=4) were presented with a brief (120 ms) four-quadrant display, each containing a texture sample. Texture samples were chosen to represent points on a line in the space of image statistics. Subjects judged which of the four texture borders was most salient. The relative salience judgments were then subjected to multidimensional scaling. In some directions, the inferred perceptual geometry matched the linear sampling of the space. But in other directions, the linear sampling of the space translated to a perceptual geometry in which opposite edges of the space were warped inward to meet each other. These findings constrain models for how the perceptual space is constructed, and, in particular, imply that opponent mechanisms alone do not suffice.